Torsion points on the boundary of the image of the canonical map

Determine whether the torsion points $(\sigma\rho)^n((-1+i\sqrt{3})/2)$ and $(\sigma\rho^{-1})^n((-1+i\sqrt{3})/2)$ lie on the boundary of the image of the Klein combination locus under a canonical map.

Background

The paper identifies torsion points that can occur in the closure of the accessible region for based fundamental domains. It conjectures that certain order-two torsion points correspond to discrete correspondences on the boundary of the Klein combination locus, and conjectures that the displayed order-three torsion points do not correspond to discrete correspondences. Their membership in the boundary of the image remains unresolved.

References

We also conjecture that the points $(\sigma\rho)n((-1+i\sqrt{3})/2)$, and $(\sigma\rho{-1})n((-1+i\sqrt{3})/2)$ cannot correspond to discrete $_a$, but it remains an open question whether these torsion points lie on the boundary of the image of $$ (Section \ref{partialK}).

— Tessellating the discreteness locus for the modular mating family of correspondences  (2608.17243 - Bullett et al., 18 Aug 2026) in Remark 2.??, Section 2, subsection “Based fundamental domains for $PSL(2,Z)$”

If we coordinatize the hyperbolic plane as the unit disc $D$, with the fixed point of $\sigma\rho\sigma$ placed at the origin in $D$, does the multi-valued function $\Psi: \setminus M_ \to D$ have the form $a \to -(f(a-1)){2/3}$, with $f$ a (single-valued) real analytic function, and is the restriction of $\Psi$ to $\Psi{cut}:{cut} \to H$ a bijection from ${cut}$ onto its image?

— Tessellating the discreteness locus for the modular mating family of correspondences  (2608.17243 - Bullett et al., 18 Aug 2026) in Section 3.3 “A possible scenario for $\Psi(^{cut})\subset H$”, first key open question